(i) An electron of mass $m$ and spin $\frac{1}{2}$ moves freely inside a cubical box of side $L$. Verify that the energy eigenstates of the system are $\phi_{l m n}(\mathbf{r}) \chi_{\pm}$where the spatial wavefunction is given by

$$\phi_{l m n}(\mathbf{r})=\left(\frac{2}{L}\right)^{3 / 2} \sin \left(\frac{l \pi x}{L}\right) \sin \left(\frac{m \pi y}{L}\right) \sin \left(\frac{n \pi z}{L}\right)$$

and

$$\chi_{+}=\left(\begin{array}{l} 1 \\ 0 \end{array}\right), \quad \chi_{-}=\left(\begin{array}{l} 0 \\ 1 \end{array}\right)$$

Give the corresponding energy eigenvalues.

A second electron is inserted into the box. Explain how the Pauli principle determines the structure of the wavefunctions associated with the lowest energy level and the first excited energy level. What are the values of the energy in these two levels and what are the corresponding degeneracies?

(ii) When the side of the box, $L$, is large, the number of eigenstates available to the electron with energy in the range $E \rightarrow E+d E$ is $\rho(E) d E$. Show that

$$\rho(E)=\frac{L^{3}}{\pi^{2} \hbar^{3}} \sqrt{2 m^{3} E}$$

A large number, $N$, of electrons are inserted into the box. Explain how the ground state is constructed and define the Fermi energy, $E_{F}$. Show that in the ground state

$$N=\frac{2}{3} \frac{L^{3}}{\pi^{2} \hbar^{3}} \sqrt{2 m^{3}}\left(E_{F}\right)^{3 / 2}$$

When a magnetic field $H$ in the $z$-direction is applied to the system, an electron with spin up acquires an additional energy $+\mu H$ and an electron with spin down an energy $-\mu H$, where $-\mu$ is the magnetic moment of the electron and $\mu>0$. Describe, for the case $E_{F}>\mu H$, the structure of the ground state of the system of $N$ electrons in the box and show that

$$N=\frac{1}{3} \frac{L^{3}}{\pi^{2} \hbar^{3}} \sqrt{2 m^{3}}\left(\left(E_{F}+\mu H\right)^{3 / 2}+\left(E_{F}-\mu H\right)^{3 / 2}\right) \text {. }$$

Calculate the induced magnetic moment, $M$, of the ground state of the system and show that for a weak magnetic field the magnetic moment is given by

$$M \approx \frac{3}{2} N \frac{\mu^{2} H}{E_{F}}$$